An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

IIT Madras Course , Prof. T.E. Venkata Balaji

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Lecture 20: Characterizing Moebius Transformations with a Single Fixed Point

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        An Introduction to Riemann Surfaces and Algebraic Curves Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit httpwww.nptel.iitm.ac.insyllabus111106044Goals To realize that in order to study Riemann surfaces with abelian fundamental group and having universal covering the upper half-plane, one needs to first classify Moebius transformations in general and in particular study among those that are automorphisms of the upper half-plane To motivate how the classification of Moebius transformations can be done using two seemingly unrelated aspects one of them being the set of fixed points in the extended complex plane and the other being the value of the square of the trace of the transformation. To show that these two aspects, though one of them is geometric while the other numeric, are in fact precisely related to each other To characterize Moebius transformations with exactly one fixed point in the extended complex plane as precisely those that are conjugate to a translation; to show that such transformations are also precisely the so-called parabolic transformations, where parabolicity is defined as the square of the trace being equal to fourKeywords Upper half-plane, unit disc, abelian fundamental group, deck transformation group, Moebius transformation, universal covering, holomorphic automorphism, group isomorphism, linear fractional transformation, bilinear transformation, fixed point of a map, square of the trace of a Moebius transformation, parabolic Moebius transformations, translations, conjugation by a Moebius transformation, special linear group, projective special linear group, upper-triangular matri

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